fq.h – finite fields¶

We represent an element of the finite field $$\mathbf{F}_{p^n} \cong \mathbf{F}_p[X]/(f(X))$$, where $$f(X) \in \mathbf{F}_p[X]$$ is a monic, irreducible polynomial of degree $$n$$, as a polynomial in $$\mathbf{F}_p[X]$$ of degree less than $$n$$. The underlying data structure is an fmpz_poly_t.

The default choice for $$f(X)$$ is the Conway polynomial for the pair $$(p,n)$$, enabled by Frank Lübeck’s data base of Conway polynomials using the _nmod_poly_conway() function. If a Conway polynomial is not available, then a random irreducible polynomial will be chosen for $$f(X)$$. Additionally, the user is able to supply their own $$f(X)$$.

Types, macros and constants¶

type fq_ctx_struct
type fq_ctx_t
type fq_struct
type fq_t

Context Management¶

void fq_ctx_init(fq_ctx_t ctx, const fmpz_t p, slong d, const char *var)

Initialises the context for prime $$p$$ and extension degree $$d$$, with name var for the generator. By default, it will try use a Conway polynomial; if one is not available, a random irreducible polynomial will be used.

Assumes that $$p$$ is a prime.

Assumes that the string var is a null-terminated string of length at least one.

int _fq_ctx_init_conway(fq_ctx_t ctx, const fmpz_t p, slong d, const char *var)

Attempts to initialise the context for prime $$p$$ and extension degree $$d$$, with name var for the generator using a Conway polynomial for the modulus.

Returns $$1$$ if the Conway polynomial is in the database for the given size and the initialization is successful; otherwise, returns $$0$$.

Assumes that $$p$$ is a prime.

Assumes that the string var is a null-terminated string of length at least one.

void fq_ctx_init_conway(fq_ctx_t ctx, const fmpz_t p, slong d, const char *var)

Initialises the context for prime $$p$$ and extension degree $$d$$, with name var for the generator using a Conway polynomial for the modulus.

Assumes that $$p$$ is a prime.

Assumes that the string var is a null-terminated string of length at least one.

void fq_ctx_init_modulus(fq_ctx_t ctx, const fmpz_mod_poly_t modulus, const fmpz_mod_ctx_t ctxp, const char *var)

Initialises the context for given modulus with name var for the generator.

Assumes that modulus is an irreducible polynomial over the finite field $$\mathbf{F}_{p}$$ in ctxp.

Assumes that the string var is a null-terminated string of length at least one.

void fq_ctx_init_randtest(fq_ctx_t ctx, flint_rand_t state, int type)

Initialises ctx to a random finite field, where the prime and degree is set according to type. To see what prime and degrees may be output, see type in _nmod_poly_conway_rand().

void fq_ctx_init_randtest_reducible(fq_ctx_t ctx, flint_rand_t state, int type)

Initializes ctx to a random extension of a prime field, where the prime and degree is set according to type. If type is $$0$$ the prime and degree may be large, else if type is $$1$$ the degree is small but the prime may be large, else if type is $$2$$ the prime is small but the degree may be large, else if type is $$3$$ both prime and degree are small.

The modulus may or may not be irreducible.

void fq_ctx_clear(fq_ctx_t ctx)

Clears all memory that has been allocated as part of the context.

const fmpz_mod_poly_struct *fq_ctx_modulus(const fq_ctx_t ctx)

Returns a pointer to the modulus in the context.

slong fq_ctx_degree(const fq_ctx_t ctx)

Returns the degree of the field extension $$[\mathbf{F}_{q} : \mathbf{F}_{p}]$$, which is equal to $$\log_{p} q$$.

const fmpz *fq_ctx_prime(const fq_ctx_t ctx)

Returns a pointer to the prime $$p$$ in the context.

void fq_ctx_order(fmpz_t f, const fq_ctx_t ctx)

Sets $$f$$ to be the size of the finite field.

int fq_ctx_fprint(FILE *file, const fq_ctx_t ctx)

Prints the context information to file. Returns 1 for a success and a negative number for an error.

void fq_ctx_print(const fq_ctx_t ctx)

Prints the context information to stdout.

Memory management¶

void fq_init(fq_t rop, const fq_ctx_t ctx)

Initialises the element rop, setting its value to $$0$$.

void fq_init2(fq_t rop, const fq_ctx_t ctx)

Initialises poly with at least enough space for it to be an element of ctx and sets it to $$0$$.

void fq_clear(fq_t rop, const fq_ctx_t ctx)

Clears the element rop.

void _fq_sparse_reduce(fmpz *R, slong lenR, const fq_ctx_t ctx)

Reduces (R, lenR) modulo the polynomial $$f$$ given by the modulus of ctx.

void _fq_dense_reduce(fmpz *R, slong lenR, const fq_ctx_t ctx)

Reduces (R, lenR) modulo the polynomial $$f$$ given by the modulus of ctx using Newton division.

void _fq_reduce(fmpz *r, slong lenR, const fq_ctx_t ctx)

Reduces (R, lenR) modulo the polynomial $$f$$ given by the modulus of ctx. Does either sparse or dense reduction based on ctx->sparse_modulus.

void fq_reduce(fq_t rop, const fq_ctx_t ctx)

Reduces the polynomial rop as an element of $$\mathbf{F}_p[X] / (f(X))$$.

Basic arithmetic¶

void fq_add(fq_t rop, const fq_t op1, const fq_t op2, const fq_ctx_t ctx)

Sets rop to the sum of op1 and op2.

void fq_sub(fq_t rop, const fq_t op1, const fq_t op2, const fq_ctx_t ctx)

Sets rop to the difference of op1 and op2.

void fq_sub_one(fq_t rop, const fq_t op1, const fq_ctx_t ctx)

Sets rop to the difference of op1 and $$1$$.

void fq_neg(fq_t rop, const fq_t op, const fq_ctx_t ctx)

Sets rop to the negative of op.

void fq_mul(fq_t rop, const fq_t op1, const fq_t op2, const fq_ctx_t ctx)

Sets rop to the product of op1 and op2, reducing the output in the given context.

void fq_mul_fmpz(fq_t rop, const fq_t op, const fmpz_t x, const fq_ctx_t ctx)

Sets rop to the product of op and $$x$$, reducing the output in the given context.

void fq_mul_si(fq_t rop, const fq_t op, slong x, const fq_ctx_t ctx)

Sets rop to the product of op and $$x$$, reducing the output in the given context.

void fq_mul_ui(fq_t rop, const fq_t op, ulong x, const fq_ctx_t ctx)

Sets rop to the product of op and $$x$$, reducing the output in the given context.

void fq_sqr(fq_t rop, const fq_t op, const fq_ctx_t ctx)

Sets rop to the square of op, reducing the output in the given context.

void fq_div(fq_t rop, const fq_t op1, const fq_t op2, const fq_ctx_t ctx)

Sets rop to the quotient of op1 and op2, reducing the output in the given context.

void _fq_inv(fmpz *rop, const fmpz *op, slong len, const fq_ctx_t ctx)

Sets (rop, d) to the inverse of the non-zero element (op, len).

void fq_inv(fq_t rop, const fq_t op, const fq_ctx_t ctx)

Sets rop to the inverse of the non-zero element op.

void fq_gcdinv(fq_t f, fq_t inv, const fq_t op, const fq_ctx_t ctx)

Sets inv to be the inverse of op modulo the modulus of ctx. If op is not invertible, then f is set to a factor of the modulus; otherwise, it is set to one.

void _fq_pow(fmpz *rop, const fmpz *op, slong len, const fmpz_t e, const fq_ctx_t ctx)

Sets (rop, 2*d-1) to (op,len) raised to the power $$e$$, reduced modulo $$f(X)$$, the modulus of ctx.

Assumes that $$e \geq 0$$ and that len is positive and at most $$d$$.

Although we require that rop provides space for $$2d - 1$$ coefficients, the output will be reduced modulo $$f(X)$$, which is a polynomial of degree $$d$$.

Does not support aliasing.

void fq_pow(fq_t rop, const fq_t op, const fmpz_t e, const fq_ctx_t ctx)

Sets rop the op raised to the power $$e$$.

Currently assumes that $$e \geq 0$$.

Note that for any input op, rop is set to $$1$$ whenever $$e = 0$$.

void fq_pow_ui(fq_t rop, const fq_t op, const ulong e, const fq_ctx_t ctx)

Sets rop the op raised to the power $$e$$.

Currently assumes that $$e \geq 0$$.

Note that for any input op, rop is set to $$1$$ whenever $$e = 0$$.

Roots¶

int fq_sqrt(fq_t rop, const fq_t op1, const fq_ctx_t ctx)

Sets rop to the square root of op1 if it is a square, and return $$1$$, otherwise return $$0$$.

void fq_pth_root(fq_t rop, const fq_t op1, const fq_ctx_t ctx)

Sets rop to a $$p^{th}$$ root root of op1. Currently, this computes the root by raising op1 to $$p^{d-1}$$ where $$d$$ is the degree of the extension.

int fq_is_square(const fq_t op, const fq_ctx_t ctx)

Return 1 if op is a square.

Output¶

int fq_fprint_pretty(FILE *file, const fq_t op, const fq_ctx_t ctx)

Prints a pretty representation of op to file.

In the current implementation, always returns $$1$$. The return code is part of the function’s signature to allow for a later implementation to return the number of characters printed or a non-positive error code.

int fq_print_pretty(const fq_t op, const fq_ctx_t ctx)

Prints a pretty representation of op to stdout.

In the current implementation, always returns $$1$$. The return code is part of the function’s signature to allow for a later implementation to return the number of characters printed or a non-positive error code.

int fq_fprint(FILE *file, const fq_t op, const fq_ctx_t ctx)

Prints a representation of op to file.

For further details on the representation used, see fmpz_mod_poly_fprint().

void fq_print(const fq_t op, const fq_ctx_t ctx)

Prints a representation of op to stdout.

For further details on the representation used, see fmpz_mod_poly_print().

char *fq_get_str(const fq_t op, const fq_ctx_t ctx)

Returns the plain FLINT string representation of the element op.

char *fq_get_str_pretty(const fq_t op, const fq_ctx_t ctx)

Returns a pretty representation of the element op using the null-terminated string x as the variable name.

Randomisation¶

void fq_randtest(fq_t rop, flint_rand_t state, const fq_ctx_t ctx)

Generates a random element of $$\mathbf{F}_q$$.

void fq_randtest_not_zero(fq_t rop, flint_rand_t state, const fq_ctx_t ctx)

Generates a random non-zero element of $$\mathbf{F}_q$$.

void fq_randtest_dense(fq_t rop, flint_rand_t state, const fq_ctx_t ctx)

Generates a random element of $$\mathbf{F}_q$$ which has an underlying polynomial with dense coefficients.

void fq_rand(fq_t rop, flint_rand_t state, const fq_ctx_t ctx)

Generates a high quality random element of $$\mathbf{F}_q$$.

void fq_rand_not_zero(fq_t rop, flint_rand_t state, const fq_ctx_t ctx)

Generates a high quality non-zero random element of $$\mathbf{F}_q$$.

Assignments and conversions¶

void fq_set(fq_t rop, const fq_t op, const fq_ctx_t ctx)

Sets rop to op.

void fq_set_si(fq_t rop, const slong x, const fq_ctx_t ctx)

Sets rop to x, considered as an element of $$\mathbf{F}_p$$.

void fq_set_ui(fq_t rop, const ulong x, const fq_ctx_t ctx)

Sets rop to x, considered as an element of $$\mathbf{F}_p$$.

void fq_set_fmpz(fq_t rop, const fmpz_t x, const fq_ctx_t ctx)

Sets rop to x, considered as an element of $$\mathbf{F}_p$$.

void fq_swap(fq_t op1, fq_t op2, const fq_ctx_t ctx)

Swaps the two elements op1 and op2.

void fq_zero(fq_t rop, const fq_ctx_t ctx)

Sets rop to zero.

void fq_one(fq_t rop, const fq_ctx_t ctx)

Sets rop to one, reduced in the given context.

void fq_gen(fq_t rop, const fq_ctx_t ctx)

Sets rop to a generator for the finite field. There is no guarantee this is a multiplicative generator of the finite field.

int fq_get_fmpz(fmpz_t rop, const fq_t op, const fq_ctx_t ctx)

If op has a lift to the integers, return $$1$$ and set rop to the lift in $$[0,p)$$. Otherwise, return $$0$$ and leave $$rop$$ undefined.

void fq_get_fmpz_poly(fmpz_poly_t a, const fq_t b, const fq_ctx_t ctx)
void fq_get_fmpz_mod_poly(fmpz_mod_poly_t a, const fq_t b, const fq_ctx_t ctx)

Set a to a representative of b in ctx. The representatives are taken in $$(\mathbb{Z}/p\mathbb{Z})[x]/h(x)$$ where $$h(x)$$ is the defining polynomial in ctx.

void fq_set_fmpz_poly(fq_t a, const fmpz_poly_t b, const fq_ctx_t ctx)
void fq_set_fmpz_mod_poly(fq_t a, const fmpz_mod_poly_t b, const fq_ctx_t ctx)

Set a to the element in ctx with representative b. The representatives are taken in $$(\mathbb{Z}/p\mathbb{Z})[x]/h(x)$$ where $$h(x)$$ is the defining polynomial in ctx.

void fq_get_fmpz_mod_mat(fmpz_mod_mat_t col, const fq_t a, const fq_ctx_t ctx)

Convert a to a column vector of length degree(ctx).

void fq_set_fmpz_mod_mat(fq_t a, const fmpz_mod_mat_t col, const fq_ctx_t ctx)

Convert a column vector col of length degree(ctx) to an element of ctx.

Comparison¶

int fq_is_zero(const fq_t op, const fq_ctx_t ctx)

Returns whether op is equal to zero.

int fq_is_one(const fq_t op, const fq_ctx_t ctx)

Returns whether op is equal to one.

int fq_equal(const fq_t op1, const fq_t op2, const fq_ctx_t ctx)

Returns whether op1 and op2 are equal.

int fq_is_invertible(const fq_t op, const fq_ctx_t ctx)

Returns whether op is an invertible element.

int fq_is_invertible_f(fq_t f, const fq_t op, const fq_ctx_t ctx)

Returns whether op is an invertible element. If it is not, then f is set of a factor of the modulus.

Special functions¶

void _fq_trace(fmpz_t rop, const fmpz *op, slong len, const fq_ctx_t ctx)

Sets rop to the trace of the non-zero element (op, len) in $$\mathbf{F}_{q}$$.

void fq_trace(fmpz_t rop, const fq_t op, const fq_ctx_t ctx)

Sets rop to the trace of op.

For an element $$a \in \mathbf{F}_q$$, multiplication by $$a$$ defines a $$\mathbf{F}_p$$-linear map on $$\mathbf{F}_q$$. We define the trace of $$a$$ as the trace of this map. Equivalently, if $$\Sigma$$ generates $$\operatorname{Gal}(\mathbf{F}_q / \mathbf{F}_p)$$ then the trace of $$a$$ is equal to $$\sum_{i=0}^{d-1} \Sigma^i (a)$$, where $$d = \log_{p} q$$.

void _fq_norm(fmpz_t rop, const fmpz *op, slong len, const fq_ctx_t ctx)

Sets rop to the norm of the non-zero element (op, len) in $$\mathbf{F}_{q}$$.

void fq_norm(fmpz_t rop, const fq_t op, const fq_ctx_t ctx)

Computes the norm of op.

For an element $$a \in \mathbf{F}_q$$, multiplication by $$a$$ defines a $$\mathbf{F}_p$$-linear map on $$\mathbf{F}_q$$. We define the norm of $$a$$ as the determinant of this map. Equivalently, if $$\Sigma$$ generates $$\operatorname{Gal}(\mathbf{F}_q / \mathbf{F}_p)$$ then the trace of $$a$$ is equal to $$\prod_{i=0}^{d-1} \Sigma^i (a)$$, where $$d = \text{dim}_{\mathbf{F}_p}(\mathbf{F}_q)$$.

Algorithm selection is automatic depending on the input.

void _fq_frobenius(fmpz *rop, const fmpz *op, slong len, slong e, const fq_ctx_t ctx)

Sets (rop, 2d-1) to the image of (op, len) under the Frobenius operator raised to the e-th power, assuming that neither op nor e are zero.

void fq_frobenius(fq_t rop, const fq_t op, slong e, const fq_ctx_t ctx)

Evaluates the homomorphism $$\Sigma^e$$ at op.

Recall that $$\mathbf{F}_q / \mathbf{F}_p$$ is Galois with Galois group $$\langle \sigma \rangle$$, which is also isomorphic to $$\mathbf{Z}/d\mathbf{Z}$$, where $$\sigma \in \operatorname{Gal}(\mathbf{F}_q/\mathbf{F}_p)$$ is the Frobenius element $$\sigma \colon x \mapsto x^p$$.

int fq_multiplicative_order(fmpz *ord, const fq_t op, const fq_ctx_t ctx)

Computes the order of op as an element of the multiplicative group of ctx.

Returns 0 if op is 0, otherwise it returns 1 if op is a generator of the multiplicative group, and -1 if it is not.

This function can also be used to check primitivity of a generator of a finite field whose defining polynomial is not primitive.

int fq_is_primitive(const fq_t op, const fq_ctx_t ctx)

Returns whether op is primitive, i.e., whether it is a generator of the multiplicative group of ctx.

Bit packing¶

void fq_bit_pack(fmpz_t f, const fq_t op, flint_bitcnt_t bit_size, const fq_ctx_t ctx)

Packs op into bitfields of size bit_size, writing the result to f.

void fq_bit_unpack(fq_t rop, const fmpz_t f, flint_bitcnt_t bit_size, const fq_ctx_t ctx)

Unpacks into rop the element with coefficients packed into fields of size bit_size as represented by the integer f.